Question: Find $c,$ given that $\lfloor c\rfloor$ satisfies \[2x^2 + 17x - 55 = 0\]and $\{ c \} = c - \lfloor c \rfloor $ satisfies \[6x^2 - 23x + 7 = 0.\]
Explanation: The first quadratic factors as \[(2x-5)(x+11) = 0,\]so its roots are $\tfrac52$ and $-11.$ Since $\lfloor c \rfloor$ must be an integer, it must be the case that $\lfloor c \rfloor = -11.$

The second quadratic factors as \[(3x-1)(2x-7) = 0,\]so its roots are $\tfrac13$ and $\tfrac72.$ Since $0 \le \{c\} < 1,$ it must be the case that $\{c\} = \tfrac13.$

Then $c = \lfloor c\rfloor + \{c\} = -11 + \tfrac13 = \boxed{-\tfrac{32}{3}}.$